Question

Two circles with radii a and b touch each other externally such that $\theta$ is the angle between the direct common tangents $(a>b\ge 2)$, then

• A. $\theta =2{\cos }^{-1}\left(\frac{a-b}{a+b}\right)$
• B. $\theta =2{\tan }^{-1}\left(\frac{a+b}{a-b}\right)$
• C. $\theta =2{\sin }^{-1}\left(\frac{a+b}{a-b}\right)$
• D. $\theta =2{\sin }^{-1}\left(\frac{a-b}{a+b}\right)$

Answer 1

The correct option is D $\theta =2{\sin }^{-1}\left(\frac{a-b}{a+b}\right)$

The line joining the point of intersection of two direct tangents with centres of circles bisect the angle.
In $△O{O}^{1}A$,
$\sin \frac{\theta }{2}=\frac{O^{1}A}{O^{1}O}=\frac{a-b}{a+b}$
$\frac{\theta }{2}={\sin }^{-1}\left[\frac{a-b}{a+b}\right]$
$\theta =2{\sin }^{-1}\left[\frac{a-b}{a+b}\right]$